



Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
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Question 20 Important Deleted for CBSE Board 2024 Exams You are here
Last updated at May 29, 2023 by Teachoo
Question 20 (Method 1) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points with position vectors π β & π β is π β = π β + π (π β β π β) Given the line passes through the points (π β β π β) = (5π Μ + 1π Μ + 6π Μ) β (3π Μ + 4π Μ + 1π Μ) = 2π Μ β 3π Μ + 5π Μ A (3, 4, 1) π β = 3π Μ + 4π Μ + π Μ B (5, 1, 6) π β = 5π Μ + 1π Μ + 6π Μ β΄ π β = (3π Μ + 4π Μ + 1π Μ) + π (2π Μ β 3π Μ + 5π Μ) Let the coordinates of the point where the line crosses the XY plane be (x, y, 0). So, π β = xπ Μ + yπ Μ + 0π Μ Since point crosses the plane, it will satisfy its equation Putting (2) in (1) xπ Μ + yπ Μ + 0π Μ = 3π Μ + 4π Μ + 1π Μ + 2ππ Μ β 3ππ Μ + 5ππ Μ xπ Μ + yπ Μ + 0π Μ = (3 + 2π)π Μ + (4 β 3π)π Μ + (1 + 5π)π Μ Two vectors are equal if their corresponding components are equal So, Solving 0 = 1 + 5π β΄ π = (βπ)/π So, x = 3 + π = 3 + 2 Γ (β1)/5 = 3 β 2/5 = 13/5 & y = 4 β 3π = 4 β 3 Γ (β1)/5 = 4 + 3/5 = 23/5 Therefore, the required coordinates are (ππ/π,ππ/π,π) Question 20 (Method 2) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points A(π₯_1, π¦_1, π§_1) and B(π₯_2, π¦_2, π§_2) is (π β π_π)/(π_π β π_π ) = (π β π_π)/(π_π β π_π ) = (π β π_π)/(π_π β π_π ) Given the line passes through the points So, the equation of line is (π₯ β 3)/(5 β 3) = (π¦ β 4)/(1 β 4) = (π§ β 1)/(6 β 1) A (3, 4, 1) β΄ π₯_1= 3, π¦_1= 4, π§_1= 1 B(5, 1, 6) β΄ π₯_2 = 5, π¦_2= 1, π§_2= 6 (π₯ β 3)/2 = (π¦ β 4)/(β3) = (π§ β 1)/5 = k So, Since, the line crosses the XY plane at (x, y, 0) z = 0 5k + 1 = 0 5k = β1 β΄ k = (βπ)/π So, x = 2k + 3 = 2 Γ (β1)/5 + 3 = 3 β 2/5 = 13/5 y = β3 Γ (β1)/5 + 4 = 4 + 3/5 = 23/5 Therefore, the coordinates of the required point are (ππ/π, ππ/π, π).